DGM: A deep learning algorithm for solving partial differential equations

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

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Nonlinear partial differential equations with applications

Progress in Nonlinear Differential Equations and their Applications

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

We study the nonlinear monotone elliptic problem  −div (a(x,∇u)) = μ on Ω, u = 0 on ∂Ω, when Ω ⊂ R , μ is a Radon measure with bounded total variation on Ω, 1 < p ≤ N , and u 7→ −div (a(x,∇u)) is a monotone operator acting on W 1,p 0 (Ω). We introduce a new definition of solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates). We also prove partial uniqueness results. SISSA, via Beirut 4, 34014 Trieste, Italy Laboratoire d’Analyse Numerique, Universite Paris VI, Case 187, 75252 Paris cedex 05, France Dipartimento di Matematica, Universita di Roma I, P.le A. Moro 2, 00185 Roma, Italy Universite d’Orleans, rue de Chartres, B.P. 6759, 45067 Orleans cedex 2, France

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Renormalized solutions of elliptic equations with general measure data

We consider the differential problem (*) { A ( u ) = μ in Ω , u = 0 on ∂ Ω , where Ω is a bounded, open subset of R N , N ≥ 2, A is a monotone operator acting on W 0 1 , p ( Ω ) , p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p -capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L 1 (Ω) and of a measure in W −1, p ′ (Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*) .

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Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Nonlinear Parabolic Equations with Measure Data

We prove existence, regularity and nonexistence results for problems whose model is 

$$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$

with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${\mathbb{R}^{N}}$. Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).

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Semilinear elliptic equations with singular nonlinearities

with Ω a bounded open subset of R , N ≥ 2, p > 1, θ ≥ 0, and f a measurable function on whose summability we will make different assumptions. It is clear from the above example that the differential operator is defined on W 0 (Ω), but that it may not be coercive on the same space as u becomes large. Due to this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied. In this paper, we will prove several existence and regularity results (depending on the summability of the datum f ) for the solutions of (1.1). Let us give the precise assumptions on the problems that we will study. Let Ω be a bounded open subset of RN , N ≥ 2. Let 1 < p < N, and let a : Ω × R × RN → RN be a Carathéodory function (that is, a(·, t, ξ) is measurable on Ω for every (t, ξ) in R × RN , and a(x, ·, ·) is continuous on R×RN for almost every x in Ω), such that the following assumptions hold: a(x, t, ξ) · ξ ≥ b(|t|) |ξ|p , (1.2)

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Luigi Orsina

Papers

Renormalized solutions of elliptic equations with general measure data

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Nonlinear Parabolic Equations with Measure Data

Semilinear elliptic equations with singular nonlinearities

Existence results for nonlinear elliptic equations with degenerate coercivity